Math 220: Summer Midterm 1 Questions

Transcription

1 Math 220: Summer 2015 Midterm 1 Questions MOST questions will either look a lot like a Homework questions This lists draws your attention to some important types of HW questions. SOME questions will have formats different than the HW (short answer, vocabulary, give an example of..., etc). This list gives examples of those. Chapters 1 and 2 since we did a lot of the same types of questions in chapters 1 and 2 (once solving them without the help of an augmented matrix and once solving them with the help of an augmented matrix), I lumped them together here. 1. Important kinds of HW questions we saw a lot Most Chap 1 and 2 questions will look some version of one of these. (a) Solve the given system of linear equations. i. Is the system consistent or inconsistent ii. Give a geometric descriptions of your answer iii. Does the system have a unique solution, infinitely many solutions, or no solutions. iv. How many solutions does the system have? v. Identify the free variables and the basic variables (b) Write this vector as a linear combination of these other vectors (or show that it s impossible) (c) Rewrite this (line or plane) from (one format) into (the other format). (d) Here are 3 points on a plane, write the equation of that plane in (ax + by + cz = d or parametric format) (e) Here is the parametric form of a (line or plane) here is the parametric for of some other (line or plane), do the two intersect? if yes, where? (f) Here s a system of linear equations with a c in it. For what values of c does the system have i. no solutions and/or ii. a unique solutions and/or iii. infinitely many solutions (g) Reduce a matrix to REF (labeling your row operations along the way) (h) Reduce a matrix to RREF (labeling your row operations along the way) (i) Here are a few planes in ax + by + cz = d format, where do they intersect? give a geometric description of the solution vectors. (j) Write a vector with norm (in the direction of/parallel to) this (vector or line) (k) It s almost a guarantee you ll have to solve a system of linear equations, and reduce a matrix to REF or RREF. 1

2 2. Vocabulary Identification (a) Are the following matrices REF? (b) Are the following matrices RREF? (c) Is the following vector a solution vector for a given system of linear equations? (d) Is the following vector a unit vector? t (e) Is x y = s z (f) Is x y z = t a line or a plane? a line or a plane? (g) Identify all pivots of the following REF matrix. 3. Give en example of each of the following (or indicate DNE) (a) A system of 2 linear equations in R 2 with a unique solution. (b) A system of 2 linear equations in R 2 with infinitely many solutions. (c) A system of 2 linear equations in R 2 with a no solutions. (d) A system of 3 linear equations in R 3 with a unique solution. (e) A system of 3 linear equations in R 3 with infinitely many solutions. (f) A system of 3 linear equations in R 3 with a no solutions. (g) A system of 2 linear equations in R 3 with a unique solution. (h) A system of 2 linear equations in R 3 with infinitely many solutions. (i) A system of 2 linear equations in R 3 with a no solutions. (j) 2 lines in R 2 that intersect in a point. (k) 2 lines in R 2 that don t intersect. (l) 3 planes in R 3 that intersect in a point. (m) 3 planes in R 3 that intersect in a line. (n) A REF matrix with an all 0 row (o) An augment matrix that corresponds to a system of linear equations that is inconsistent (p) An augment matrix that corresponds to a system of linear equations that is consistent (q) An augment matrix that corresponds to a system of linear equations that has infinity many solutions (r) An augment matrix that corresponds to a system of linear equations that has a unique solution 2

3 4. Is Format unique? (a) Is the parametric form[ of ] a line[ in R] 2 unique? [ ] If not give a different parametric for of the line = t + in x = tu + v format. x 2 1 y 3 4 (b) Is the parametric form of a line in R 3 unique? If not give a different parametric for of the line x y = t in x = tu + v format. z 3 4 (c) Is the parametric form of a plane in R 3 unique? If not give adifferent parametric for of the plane su + tv + w format. x y z = s t in x = (d) Is the ax + by = c format for of a line in R 2 unique? If not give a different description of the line the line 3x 2y = 17 in ax + by = c format. (e) Is the ax + by + cz = d format for of a plane in R 3 unique? If not give a different description of the line the line 2x 1 y+3z = 2 in ax+by+cz = d 2 format. (f) Is the REF of a matrix unique? If not give another another REF matrix that is equivalent to (g) Is the RREF of a matrix unique? Ifnot give another another RREF matrix that is equivalent to Indicate if the following statements are true or false: (a) Any T/F from chapters 1 and 2. (b) All systems of linear equations have at least one solution. (c) All systems of linear equations are consistent. (d) All homogeneous systems of linear equations have at least one solution. (e) All homogeneous systems of linear equations are consistent. (f) All systems of 2 linear equations in 3 unknowns have infinitely many solutions. (g) All consistent systems of 2 linear equations in 3 unknowns have infinitely many solutions. 3

4 (h) All homogeneous systems of 2 linear equations in 3 unknowns have infinitely many solutions. (i) There exist a system of 2 linear equations in 3 unknowns that has a unique solution. (j) There exist a homogeneous system of 2 linear equations in 3 unknowns that has a unique solution. (k) All systems of 3 linear equations in 3 unknowns are inconsistent. (l) All systems of 3 linear equations in 3 unknowns are a unique solution. (m) All systems of 3 linear equations in 3 unknowns have infinitely many solutions. (n) Let Ax = b be a system of linear equations. And let A be an m n matrix. i. If the rank of A = n, then there is a unique solution. ii. If the rank of A = n and the system is consistent, then there is a unique solution. iii. If the rank of A < n, then there are infinitely many solutions. iv. If the rank of A < n, and the system is inconsistent, then there are infinitely many solutions. (o) Every 3 3 matrix is row equivalent to the matrix Chapter 3 6. Important kinds of HW questions Most Chap 3 questions will look some version of one of these. (a) Here are some matrices, add, subtract, scalar multiply, matrix multiply, raise them to some power, and/or transpose them in some way. (Questions 1, 2, 3, 15 in chapter 3) (b) Find the inverse of some Elementary Matrix (c) Here s a matrix A, find A 1 (if it exists) i. use A 1 to solve some system of linear equations given in terms of A. (d) Solving an equation involving matrices (like questions 5, 6 in chap 3) (e) Something cute about inverses (like 21-23) (f) Any prove or prove or give a counter example questions will be directly off the assigned HW. (g) It s almost a guarantee you ll have to multiply matrices, and find an inverse 4

5 7. Vocabulary Identification: (a) Is the following matrix Upper Triangular, Strict Upper Triangular, Lower Triangular, Strict Lower Triangular (b) Is the following matrix diagonal? (c) Is the following matrix Symmetric, antisymmetric, or neither? (d) Is the following matrix singular or non-singular? (e) For matrix A given, what is (A) 25 (f) Is the following matrix an Identity Matrix? (g) Is the following an Elementary Matrix? (h) Is the following a square matrix? 8. Random (a) Give an example of 2 matrices A, B so that AB = BA (b) Give an example of 2 matrices A, B so that AB BA (c) Find c and d so that B = 5 2 c 2 d 5 is the inverse of A = Determine if the following are True or False: (a) Any T/F from chapter 3 (b) True / False: For all matrices A, B, AB = BA (c) True / False: For all n n matrices, AB = BA (d) True / False: Matrices must be the same size to be added (e) True / False: Matrices must be the same size to be multiplied (f) True / False: For all matrices, (A t ) t = A. (g) True / False: For all n n matrices, (A t ) t = A. (h) True / False: For A, B m n matrices and C an n r matrix, if A = B, then AC = BC (i) True / False: For A, B m n matrices and C an n r matrix, if AC = BC, then A = B (j) True / False: For A an m n matrix and B an n r matrix (AB) t = A t B t (k) True / False: For A an m n matrix and B an n r matrix (AB) t = B t A t (l) True / False: For A any n n matrix, A t = A. (m) True / False: For any square matrix A, and scalar c, (ca) t = c(a t ) (n) True / False: For any square matrix A, and scalar c, (ca) 1 = c(a 1 ) (o) True / False: Every matrix has an inverse. (p) True / False: Every square matrix has an inverse. (q) True / False: The product of two n n upper triangular matrices is an n n upper triangular matrix. 5

6 (r) True / False: The product of two n n diagonal matrices is an n n diagonal matrix. (s) True / False: For any matrix A, the matrix B = A t A is antisymmetric. (t) True / False: For any matrix A, the matrix B = A t A is symmetric. (u) True / False: For any matrix A, the matrix B = A t A is diagonal. (v) True / False: For any matrix A, the matrix B = A t A has an inverse. 6